The Complex Exponential Function

Recall that, for real-valuedx the exponential function ex, also denoted by exp(x), has the series representation

The property ex+y = ex ey corresponds to the equality

k=0

^ (x + y)k

k=0 k!

= E 1 Z(k)xk—mym

k! m

k= 0 m = 0

Ж k xk—m ym

The first equality in (Ш.11) results from the binomial expansion, and the last equality follows easily by rearranging the summation. It is easy to see that (Ш.11) also holds for complex-valued x and y. Therefore, we can define the complex exponential function by the series expansion (Ш.10):

(III.12)

Moreover, it follows from Taylor’s theorem that

о / nm i2m со / nm u2„ +1

cose») _E (-12_.^, Srn(b) _£ (-1>, + . (Ш.13)

and thus (III.12) becomes

ga+ib _ ga [cos(b) + i • sin(b)].

Setting a _ 0, we find that the latter equality yields the following expressions for the cosines and sine in terms of the complex exponential function: